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\title{Math 567 Fall 2008 Number Theory I, Problems 6}\endtitle
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\centerline{\bf To be submitted by Tuesday, October 7th}
\medskip
\noindent {\bf Easier problems}
\settabs\+aaaa&aaaaa&aaaaaaaaaaaaa\cr
\medskip
\noindent 1. Show that $\left(
\sum_{m|n}d(m)
\right)^2=\sum_{m|n}d(m)^3$.
\medskip
\noindent 2. Show that if $\sigma(n)$ is odd, then $n$ is a square or twice a %%@
square.
\medskip
\noindent 3. Show that $\sum_{l|(m,n)}\mu(l)$ is $1$ when $(m,n)=1$ and is $0$ %%@
otherwise. Hence prove that $\sum_{m=1;(m,n)=1}^nm=\frac12n\phi(n)\quad %%@
\text{when}\quad n>1$.
\medskip
\noindent 4. Let $\lambda(n)=(-1)^{\Omega(n)}$ (Liouville's function). Show that %%@
$\lambda(n)=\sum_{m^2|n}\mu\left(
n/m^2
\right)$.
\medskip
\noindent 5. Define $f(n)$ to be $(-1)^{\frac{n-1}{2}}$ when $n$ is odd, $0$ when %%@
$n$ is even. Show that $f$ is totally multiplicative and is periodic with period %%@
$4$.
\medskip
\noindent {\bf Harder problem}
\medskip
\+6.&Let $k\in{\Bbb N}$, $z\in{\Bbb C}$, $e(\alpha)=\exp(2\pi i\alpha)$. Define %%@
$\Phi_k(z)=\prod_{l=1;(l,k)=1}^k(z-e(l/k))$,\cr
\noindent the $k$-th cyclotomic polynomial, i.e. the monic polynomial whose roots %%@
are the primitive $k$-th roots of unity.
\+&(i)&Show that $\prod_{l|k}\Phi_l(z)=z^k-1$ and $\Phi_1(z)=z-1$.\cr
\+&(ii)&Deduce that $\Phi_k(z)=\prod_{l|k}\left(
z^l-1
\right)^{\mu(k/l)}$.\cr
\+&(iii)&Show that if $k>1$, then $\Phi_k(z)=\prod_{l|k}\left(
1-z^l
\right)^{\mu(k/l)}$ and $\Phi_k(0)=1$.\cr
\+&(iv)&By considering the expansion $(1-z^l)^{-1}=1+z^l+z^{2l}+\cdots$ when %%@
$|z|<1$\cr
\noindent show that $\Phi_k(z)$ has integer coefficients.
\+&(v)&Let $K$ be the largest squarefree divisor of $k$. Show that %%@
$\Phi_k(z)=\Phi_K(z^{k/K})$.\cr
\+&(vi)&Prove that $\Phi_p(z)=1+z+\cdots+z^{p-1}$.\cr
\+&(vii)&Show that if $k$ is odd and $k>1$, then $\Phi_{2^rk}(z)=\Phi_k\left(
-z^{2^{r-1}}
\right)$.\cr
\+&(viii)&Suppose that $p$ and $q$ are different primes. Show that, when $|z|<1$, %%@
$\Phi_{pq}(z)$\cr
\noindent $=(1-z)\sum_{n=0}^{\infty}b_nz^n$ where $b_n$ is the number of choices of %%@
$u,v\in{\Bbb Z}$ with $0\le u\le q-1$, $v\ge0$ and $up+vq=n$. Deduce that $b_n=0$ %%@
or $1$ and that the coefficients of $\Phi_{pq}(z)$ are $\pm1$ or $0$.\par
\+&(ix)&Show that if $k<105$, then the coefficients of $\Phi_k(z)$ are $\pm1$ or %%@
$0$.\cr
\+&(x)&Show that the coefficient of $z^7$ in $\Phi_{105}$ is $-2$. It is known %%@
(Erd\H os 1948,\cr
\noindent Vaughan 1975) that {\it sometimes} the coefficients of $\Phi_k(z)$ are as %%@
large as $\exp\left(
2^{\frac{\log k}{\log\log k}}
\right)$ and that ``almost always'' the largest coefficient is arbitrarily large %%@
(Meier 1995). So much for intuition ...!\par
\+&(xi)&Prove that if $k>1$, then $\Phi_k(1)=e^{\Lambda(k)}$.\cr
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